A table has a sinusoidal vibration with constant w Which is

A table has a sinusoidal vibration with constant w. Which is the biggest movement it can present, if a rigid object of mass M must mantain contact with the table at all moments?

Solution

The contact will be lost when acceleration due to vibration is more than acceleration due to gravity. Hence for all time maintaining the contact between table and mass,acceleration due to vibration should be always less or equal to acceleration due to gravity. i.e.
A_v<=A_g

Where A_v = X\'\'(t) = - X_oW² Sin(Wt)

So, -X_oW² Sin(Wt)<=g

or X(t) W² <= g

considering maximum case,

X(t) W² = g

X(t)_max = g/W² ... Ans.